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FunctionEnglishPPTcourseware目录FunctionIntroductionThebasicformoffunctionsOperationoffunctionsImageoffunctionThepracticalapplicationoffunctions01FunctionIntroductionFunctionDefinitionAfunctionisafundamentalconceptinmathematics,whichisamappingrelationshipfrominputtooutput.Functiondefinitionstypicallyincludeinputsandoutputs,aswellastheirrelationships.FunctionexpressionAfunctioncanbeexpressedusingmathematicalexpressions,andtheformoftheexpressiondependsonthetypeoffunction.Forexample,linearfunctions,polynomialfunctions,trigonometricfunctions,etc.allhavetheirownspecificexpressions.DefinitionoffunctionsCertainty01Thefunctionalrelationshipisdeterministic,andforeachinputvalue,thereisauniqueoutputvaluecorrespondingtoit.Continuity02Withinacertaininterval,thegraphofafunctioniscontinuous,meaningthatthevalueofthefunctiondoesnotexperienceanydiscontinuityorjump.Computability03Foragiveninputvalue,thefunctionisalwaysabletocalculateauniqueoutputvalue.CharacteristicsoffunctionsClassificationoffunctionsAlgebraicfunctionsFunctionsrepresentedbyalgebraicexpressions,suchaspolynomialfunctions,fractionalfunctions,etc.Trigonometricfunctionsfunctionsrelatedtotrigonometry,suchassinefunctions,cosinefunctions,tangentfunctions,etc.Exponentialfunctionafunctionofexponentialgrowth,suchasnaturalexponentialfunction,powerfunction,etc.Transcendentfunctionafunctionthatcannotberepresentedbyalgebraicmethods,suchaslogarithmicfunctions,hyperbolicfunctions,etc.02ThebasicformoffunctionsSummaryLinearfunctionisthesimplestformoffunction,expressedasy=kx+b,wherekandbareconstants.要点一要点二DetaileddescriptionAlinearfunctionrepresentsalinearrelationship,wherethereisafixedslopekandinterceptbonthey-axisbetweenxandy.Linearfunctionshaveawiderangeofapplicationsinreallife,suchastherelationshipbetweenspeedandtime,andtherelationshipbetweencostandquantity.LinearfunctionSummary:Thepowerfunctionisrepresentedasy=x^n,wherenisaconstant.Detaileddescription:Thepowerfunctionrepresentsthenthpowerofx,anditsshapedependsonthevalueoftheexponentn.Whenn>0,thegraphofthefunctionismonotonicallyincreasing;Whenn<0,thegraphofthefunctionismonotonicallydecreasing.Powerfunctionsareveryusefulindescribingpracticalphenomenasuchaspopulationgrowth,radioactivedecay,etc.PowerfunctionTheexponentialfunctionisrepresentedasy=a^x,whereaisanormalnumber.SummaryTheexponentialfunctionrepresentsthepowerofbaseatothex,anditsgrowthrateaccelerateswiththeincreaseofx.Exponentialfunctionsareveryusefulindescribingpracticalproblems,suchascompoundinterestcalculations,populationgrowthforecasts,etc.DetaileddescriptionExponentialfunctionThelogarithmicfunctionisexpressedasy=log(x),wherex>0.SummaryThelogarithmicfunctionrepresentsthelogarithmofanumber,thatis,thelogarithmbasedon10ore.Logarithmicfunctionsareveryusefulinsolvingpracticalproblems,suchasmeasuringsoundintensityandevaluatingearthquakemagnitude.DetaileddescriptionLogarithmicfunction03OperationoffunctionsAdditionoffunctionsAddtwofunctionstogenerateanewfunction.SummaryFunctionadditionreferstoaddingthecorrespondingoutputvaluesoftwofunctionstogenerateanewfunction.Theinputvalueofthisnewfunctionisthesameastheinputvalueoftheoriginalfunction,andtheoutputvalueisthesumoftheoutputvaluesoftheoriginalfunction.DetaileddescriptionSummarySubtractonefunctionfromanothertogenerateanewfunction.DetaileddescriptionFunctionsubtractionreferstosubtractingtheoutputvalueofanotherfunctionfromthecorrespondingoutputvalueofonefunctiontogenerateanewfunction.Theinputvalueofthisnewfunctionisthesameastheinputvalueoftheoriginalfunction,andtheoutputvalueisthedifferencebetweentheoutputvalueoftheoriginalfunction.SubtractionoffunctionsMultiplytwofunctionstogenerateanewfunction.SummaryFunctionmultiplicationreferstomultiplyingtheoutputvaluesoftwofunctionstogenerateanewfunction.Theinputvalueofthisnewfunctionisthesameastheinputvalueoftheoriginalfunction,andtheoutputvalueistheproductoftheoutputvalueoftheoriginalfunction.DetaileddescriptionMultiplicationoffunctionsSummaryDivideonefunctionbyanothertogenerateanewfunction.DetaileddescriptionFunctiondivisionreferstodividingtheoutputvalueofonefunctionbytheoutputvalueofanotherfunctiontogenerateanewfunction.Theinputvalueofthisnewfunctionisthesameastheinputvalueoftheoriginalfunction,andtheoutputvalueisthequotientoftheoutputvalueoftheoriginalfunction.Divisionoffunctions04ImageoffunctionDrawingthefunctiongraphUsethegraphicstoolinthePPTtoplotthefunction'sinputandoutputvaluesontheCartesianplaneThishelpsvisualizethebehaviorofthefunctionGraphannotationsAddlabels,axestitles,andotherannotationstoclarifythegraph'smeaningandpurposeThedrawingoffunctiongraphs

TransformationoffunctionimagesHorizontalscalingAdjustthex-axisscaletohighlightspecificfeaturesoremphasisinregionsofthefunction'sbehaviorVerticalscalingAdjustthey-axisscaletoemphasisordeemphasisthefunction'soutputvaluesTranslationShiftthegraphalongthex-oryaxistocenteritaroundaspecificpointortocompareitwithotherfunctionsIdentifyingcriticalpointsIdentifyanycriticalpoints(extreme)onthegraph,suchaslocalmaximaandminima,smallpoints,andpointsofinflationDerivativeanalysisUsethederivativeofthefunctiontoanalyzeitsslopeandcurvatureatvariouspointsonthegraphInterpretationofgraphfeaturesInterpretthesignificanceoffeaturesonthegraph,assymbols,discontinuities,andregionsofincreasingordecreasingbehaviorAnalysisofFunctionImages05ThepracticalapplicationoffunctionsApplicationinPhysicsModelingMechanicalPhenomena:Functionscanbeusedtomodelandanalyzemechanicalsystems,suchassprings,pendulums,andvibrationsTheyallowforthepredictionofmotionpatterns,understandingofenergytransfer,andoptimizationofsystemperformanceElectricalTheory:FunctionsplayaclinicalroleinthedescriptionandanalysisofelectricalfieldsTheyareusedtoreportwaveforms,describefieldpatterns,andcalculateinteractionsbetweenchargesandcurrentsThermodynamics:Inthermodynamics,functionsareemployedtoreportthermodynamicprocesses,analyzephasetransitions,andcalculatethermodynamicpropertiessuchasentropy,enthalpy,andfreeenergyApplicationintheeconomyEconomicModeling:Functionsareusedineconomicmodelingtoreportrelationshipsbetweenvariablessuchassupplyanddemand,productionandconsumption,andpricesandquantitiesTheyprovideinsightsintomarketbehaviorandtrendsFinancialAnalysis:Functionsareessentialinfinancialanalysis,specificallyintheevaluationofinvestmentreturns,portfoliooptimization,andriskassessmentTheyallowforthecalculationofreturns,evaluationofassetprices,andconstructionoffinancialindicatorsBusinessDecisionMaking:Functionscanbeutilizedinvariousbusinessdecisions,suchasdemandforecasting,productionplanning,a